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SQA Higher Maths Integration of exponentials and logarithms

Integration of exponentials and logarithms

This page covers the following topics:

To find the integral of equations or variables in the form eΛ£, use the following rule.

Integrating basic exponent

Integrate variables in the form aΛ£ using the exponentials rule. Integrate 2 functions multiplied together by parts; for u, choose the function that is simpler when differentiated.

Integrating exponents

Integrate variables in the form lnx using integration by parts: ∫ u (dv/dx) dx = uv βˆ’ ∫ v (du/dx) dx. Let dv/dx = 1 and u = the variable in the form lnx.

Integrating natural logarithm

Integrate logarithmic functions using integration by parts: ∫ u (dv/dx) dx = uv βˆ’ ∫ v (du/dx) dx. Let dv/dx = 1 and u = the logarithmic function.

Integrating logarithms

1

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Find ∫ e³ˣ ⁻ ¹ dx.

SQA Higher Maths Integration of exponentials and logarithms Find ∫ e³ˣ ⁻ ¹ dx.
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2

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Integrate (eΛ£ βˆ’ 2)Β²/eΒ²Λ£ dx.

SQA Higher Maths Integration of exponentials and logarithms Integrate (eΛ£ βˆ’ 2)Β²/eΒ²Λ£ dx.
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3

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Integrate 2Λ£ dx.

SQA Higher Maths Integration of exponentials and logarithms Integrate 2Λ£ dx.
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4

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Find ∫ xe²ˣ dx.

SQA Higher Maths Integration of exponentials and logarithms Find ∫ xe²ˣ dx.
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5

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The diagram shows part of the curve y = βˆ’4Β²Λ£ ⁻ ⁸ + 5. Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the xβˆ’axis.

SQA Higher Maths Integration of exponentials and logarithms The diagram shows part of the curve y = βˆ’4Β²Λ£ ⁻ ⁸ + 5. Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the xβˆ’axis.
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6

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Find the integral ∫ 3lnx dx.

SQA Higher Maths Integration of exponentials and logarithms Find the integral ∫ 3lnx dx.
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7

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The diagram shows part of the curve y = βˆ’ln(x + 1) βˆ’ 2xΒ² + 8. Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the xβˆ’axis.

SQA Higher Maths Integration of exponentials and logarithms The diagram shows part of the curve y = βˆ’ln(x + 1) βˆ’ 2xΒ² + 8. Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the xβˆ’axis.
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8

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Integrate ∫ x(lnx)² dx.

SQA Higher Maths Integration of exponentials and logarithms Integrate ∫ x(lnx)² dx.
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9

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Integrate xlogβ‚…(4x) dx.

SQA Higher Maths Integration of exponentials and logarithms Integrate xlogβ‚…(4x) dx.
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10

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What is ∫ log₃(4x + 8) dx?

SQA Higher Maths Integration of exponentials and logarithms What is ∫ log₃(4x + 8) dx?
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